Abstract
We prove that for analytic functions in low dimension, the convergence rate of the deep neural network approximation is exponential.
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References
Barron A R. Universal approximation bounds for superpositions of a sigmoidal function. IEEE Trans Inform Theory, 1993, 39: 930–945
Cybenko G. Approximation by superpositions of a sigmoidal function. Math Control Signal Syst, 1989, 2: 303–314
Liang S, Srikant R. Why deep neural networks for function approximation? ArXiv:1610.04161, 2016
Lu Z, Pu H, Wang F, et al. The expressive power of neural networks: A view from the width. In: Advances in Neural Information Processing Systems, vol. 30. Long Beach: Neural Information Processing Systems Foundation, 2017, 6232–6240
Montanelli H, Du Q. Deep ReLU networks lessen the curse of dimensionality. ArXiv:1712.08688, 2017
Poggio T, Mhaskar H, Rosasco L, et al. Why and when can deep-but not shallow-networks avoid the curse of dimen-sionality: A review. Internat J Automat Comput, 2017, 14: 503–519
Telgarsky M. Benefits of depth in neural networks. ArXiv:1602.04485, 2016
Yarotsky D. Error bounds for approximations with deep relu networks. Neural Networks, 2017, 94: 103–114
Acknowledgements
This work was supported by Offce of Naval Research (ONR) (Grant No. N00014-13-1-0338) and Major Program of National Natural Science Foundation of China (Grant No. 91130005). The authors are grateful to Chao Ma for very helpful discussions during the early stage of this work. The authors are also grateful to Jinchao Xu for his interest, which motivated us to write up this paper.
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Dedicated to Professor TaTsien Li on the Occasion of His 80th Birthday
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E, W., Wang, Q. Exponential convergence of the deep neural network approximation for analytic functions. Sci. China Math. 61, 1733–1740 (2018). https://doi.org/10.1007/s11425-018-9387-x
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DOI: https://doi.org/10.1007/s11425-018-9387-x